Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. In A.Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.
The research project described in this article strives to understand how students with little or no statistical knowledge begin to understand informal inference. Their interdisciplinary study of fifth grade students at a magnet primary school in Haifa focuses on the use of real data and technology to develop statistical reasoning. Students analyzed data from the UK CensusAtSchool database using TinkerPlots and began making informal inferences based on their analyses. Students were asked if these inferences would hold for larger samples and to justify their reasoning. At the same time, students investigated systems in their science curriculum that they could also describe with statistics. From this research, a learning trajectory was created to support the development of informal inference; preliminary results reported that this trajectory seems to be effective. At the time the article was written, the final results were not available.
Burrill, G. F. & Elliott, P. C. (2006). Preface. In G. F. Burrill & P. C. Elliott (Eds.), Thinking and Reasoning with Data and Chance: Sixty-eighth Yearbook (pp. ix-xii). Reston: VA: National Council of Teachers of Mathematics.
In the preface to this NCTM Yearbook, Burrill and Elliott provide an overview of the yearbook and the scope of the topics addressed in the collection of articles. They provide a brief history of NCTM publications that address statistics and probability, highlighting the first NCTM yearbook on statistics and probability published in 1981 and the NCTM Curriculum and Evaluation Standards for School Mathematics published in 1989. Burrill and Elliott then briefly compare the 1981 NCTM yearbook with this one published in 2006 and emphasize the prominent role that technology plays in teaching and learning statistics. They close with a summary of the articles included in the yearbook and acknowledgements of the authors and contributors.
Garfield, J., & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of research on teaching and learning statistics. International StatisticalReview,75(3), 372-396.
Garfield published a paper reviewing research on teaching and learning statistics in 1995. In the years that passed between the date of that publication and the date of this article, much new research had been performed on the subject. Garfield and Ben-Zvi “revisit” Garfield’s original contribution in 1995 to highlight new developments and discoveries that emerged from the research conducted between 1995 and the publication of this article. The literature review includes research from multiple disciplines, including mathematics education and psychology, among others, and provides implications for statistics instruction and future research on teaching foundational concepts, such as variability and inference. The paper is organized into four sections: A review of recent research on teaching and learning statistics; a summary of a shifted focus in the research community on the differences between “statistical literacy, reasoning and thinking”; research on building students’ reasoning skills about distribution, center and variability; and implications for future research on teaching statistics.
Garfield, J., & Ben-Zvi, D. (2008) Developing students’ statistical reasoning:Connecting research and teaching practice. New York: Springer.
Chapter 13 of this book is titled “Learning to reason about statistical inference.” The chapter discussed the importance of understanding statistical inference, its place in the curriculum, difficulties and misconceptions, teacher’s understanding, students’ dispositions and implications from research. The chapter also identifies instructional tasks that foster development of skills in statistical inference and the ideas and concepts that students should gain through engagement with the tasks. The chapter supports the use of resampling tasks to teach informal inference as well as probability simulations. Finally, the chapter provides suggestions on teaching statistical inference.
Marzano, R. J. (2010). Teaching inference. Educational Leadership, 67(7), 80-81.
In this article, Marzano argues that the process of inference is critical for students to engage in as they develop their critical thinking and reasoning skills. He provides key questions for teachers to ask to elicit inferential responses from their students and promote critical thinking in the classroom. The first question, “What is my inference?”, is designed to help students identify what conclusions or predictions they have made in their statistical process. The second question, “What information did I use to make this inference?”, attempts to help students realize what information they used in their reasoning toward their conclusions. The third question, “How good is my thinking?”, students are guided to analyze whether their process and investigation are valid and that their conclusions are based on strong evidence. The final question Marzano encourages teachers to pose is “Do I need to change my thinking?”, which is designed to build students’ critical thinking skills and attention to new information that may require revision of their original inference. In summary, Marzano presents tools for teachers to help their students develop their skills in inference by asking targeted questions and engaging in critical analysis of their own reasoning.
Paparistodemou, E., & Meletiou-Mavrotheris, M. (2008). Developing youngstudents’ informal inference skills in data analysis.Statistics EducationResearch Journal.7(2). 83-106.
In this paper, the authors describe a study where third graders created a survey for students at their school to collect and analyze relevant data. The study suggests that by using data that is interesting and relatable to students, inferential reasoning can begin to develop. The authors also found that using TinkerPlots further enhances students’ development of inferential reasoning by providing an easy way to dynamically view the data and test conjectures. Students in this study were highly motivated and, interestingly, began making data-based arguments on their own. After providing initial support for a conjecture, researchers prompted the students to consider applying the conjecture to a larger population and begin thinking about the ideas of chance. The authors encourage early statistical instruction to be focused on an open-ended approach driven by data where students can begin to reason informally rather than through teaching procedures related to inference. By focusing on an informal approach, the authors argue that students will have time to think through the data and make sense of what is happening in context, thus laying the foundation for a better understanding of formal inferential reasoning later.
Rossman, A. J. (2008). Reasoning about informal statistical inference: One statistician’sview.Statistics Education Research Journal. 7(2). 5-19.
In this article, Rossman contends that statistical inference is what is done after data has been collected. Inference involves the interpretations, implications and insights that follow observations about the data. Rossman presents a comparison between formal and informal inference teaching models and provides a number of examples of activities (specifically randomization tests) to illustrate how inference can be taught in the classroom. Rossman argues that using probability and simulations (repeated sampling, randomization tests) is a powerful way to introduce students to engage in statistical inference and reasoning.
Rossman, A. J., & Chance, B. L. (1999). Teaching the reasoning of statistical inference: A"top ten" list. The College Mathematics Journal. 30(4), 297-305.
In this article, Rossman and Chance argue that critical principles of instruction that include the promotion of “active learning” and work with the practical application of statistics are missing from current instruction of statistical inference. They offer a list of ten key practices that teachers should engage in when teaching statistical inference. For each of these practices, the authors provide examples to illustrate the principles underlying each practice. In order to help students develop inferential reasoning, the Rossman and Chance encourage teachers to provide opportunities for students to perform physical simulations to “make sense” of the data they collect, use technology to enhance exploration of phenomena, and use robust statistical measures, such as tests for significance and confidence intervals, to ensure strength of their interpretations of the data. Rossman and Chance also encourage teachers to emphasize that certain representations and methods of collection can misrepresent the data and result in misinterpretations and false conclusions. Finally, the authors strongly encourage teachers to provide students with frequent opportunities to make connections between their data and the real world and communicate their reasoning through discussion and in written form.
Stohl, H., & Tarr. J. E. (2002). Developing notions of inference using probability simulationtools. Journal of Mathematical Behavior, 21. 319-337.
This article describes the interactions between two sixth-grade students during a twelve day teaching experiment where the students used Probability Explorer software. Six problem-based tasks were used during the teaching experiment to develop students’ reasoning of probability and concept of inference. The tasks required students to model situations, collect, display and analyze data, and generate convincing arguments. As students progressed through the tasks, they developed a foundation for understanding the power of larger samples. The authors also found that interactions with microworld tools were crucial in developing students’ reasoning and combating their misconceptions.
Tarr, J. E., Stohl Lee, H., & Rider, R. L. (2006). When data and chance collide: drawinginference from empirical data.In G. F. Burrill & P. C. Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139 – 150). Reston, VA: National Council of Teachers of Mathematics.
This chapter discusses a task used with sixth-grade students where they learned to draw inferences from empirical data. The task was called “Schoolopoly”, and students were assigned to test if the dice they were using were “fair” and support their decision with compelling evidence. The task was completed using Probability Explorer, TI graphing calculator or by hand. In summary, students had rich discussions that supported the idea that sample size plays a critical role in statistical inference.
Watson, J. M. (2001). Longitudinal development of inferential reasoning by schoolstudents. Educational Studies in Mathematics, 47(3). 337-372.
This article describes follow-up interviews of students in Australia who were involved in the author's previous study (Watson, J.M. & Moritz, J. B., 1999) where students were asked to compare two data sets in graphical form. The author found that in the later interviews, the increase in the outcome of the students’ responses may have been due to developmental changes rather than explicit teaching related to these concepts. Thus, the author reminds teachers that some students may need more support when dealing with analyzing unequal data sets, while other students may be ready to move forward. The author’s findings in these interviews suggest that students in higher grade levels are developmentally more equipped to provide more sophisticated conjectures.
Watson, J. M., & Moritz, J. B. (1999) The beginning of statistical inference: Comparingtwo data sets. Educational Studies in Mathematics.37(2), 145-168.
This article describes a study of third to ninth grade students in Australia who were interviewed to assess their understanding when comparing two data sets in graphical form. There were four data sets presented, one set at a time, each increasing in difficulty. As the students reasoned through the problem, interviewers took notes and ranked their responses on a scale describing the level of the students’ responses. They found, that in general, students’ in higher grade levels gave more meaningful responses and descriptions of the data.
Wild, C. J., Pfannkuch, M., Regan, M., & Horton, N. J. (2011). Towards moreaccessible conceptions of statistical inference.Journal of the RoyalStatistical Society. Series A, 174, 247-295.
This article presents a case for intentionally laying a foundation for statistical inference using visual clues. Through computer animations simulating repeated sampling, students can begin to visualize the effects sample size have on variation and, thus, begin to make connections needed to make comparisons leading to inference. The idea is that students will better understand measures of center and can start to make informal inferences by looking at the spread of data as depicted on a box plot. As students progress through the author’s “milestones”, they continue the progression towards better understanding how to make formal inferences. The authors argue that this less complex and concrete foundation is essential for students to begin truly understanding statistical inference.
Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2008). A framework to support research on informal inferential reasoning. Statistics Education ResearchJournal. 7(2), 40-58.
This article synthesizes research of how informal knowledge and informal reasoning has been defined and how they relate to drawing inferences in statistics. Based on the research, the authors propose a working definition of Inferential Informal Reasoning (IIR) and seek to provide a framework which can help design and evaluate tasks that elicit IIR. Their hope is that this framework will inform and inspire further research and a refined definition of IIR in the future.
Back to Inference Home
Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. In A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.
The research project described in this article strives to understand how students with little or no statistical knowledge begin to understand informal inference. Their interdisciplinary study of fifth grade students at a magnet primary school in Haifa focuses on the use of real data and technology to develop statistical reasoning. Students analyzed data from the UK CensusAtSchool database using TinkerPlots and began making informal inferences based on their analyses. Students were asked if these inferences would hold for larger samples and to justify their reasoning. At the same time, students investigated systems in their science curriculum that they could also describe with statistics. From this research, a learning trajectory was created to support the development of informal inference; preliminary results reported that this trajectory seems to be effective. At the time the article was written, the final results were not available.
Burrill, G. F. & Elliott, P. C. (2006). Preface. In G. F. Burrill & P. C. Elliott (Eds.), Thinking and Reasoning with Data and Chance: Sixty-eighth Yearbook (pp. ix-xii). Reston: VA: National Council of Teachers of Mathematics.
In the preface to this NCTM Yearbook, Burrill and Elliott provide an overview of the yearbook and the scope of the topics addressed in the collection of articles. They provide a brief history of NCTM publications that address statistics and probability, highlighting the first NCTM yearbook on statistics and probability published in 1981 and the NCTM Curriculum and Evaluation Standards for School Mathematics published in 1989. Burrill and Elliott then briefly compare the 1981 NCTM yearbook with this one published in 2006 and emphasize the prominent role that technology plays in teaching and learning statistics. They close with a summary of the articles included in the yearbook and acknowledgements of the authors and contributors.
Garfield, J., & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of research on teaching and learning statistics. International Statistical Review, 75(3), 372-396.
Garfield published a paper reviewing research on teaching and learning statistics in 1995. In the years that passed between the date of that publication and the date of this article, much new research had been performed on the subject. Garfield and Ben-Zvi “revisit” Garfield’s original contribution in 1995 to highlight new developments and discoveries that emerged from the research conducted between 1995 and the publication of this article. The literature review includes research from multiple disciplines, including mathematics education and psychology, among others, and provides implications for statistics instruction and future research on teaching foundational concepts, such as variability and inference. The paper is organized into four sections: A review of recent research on teaching and learning statistics; a summary of a shifted focus in the research community on the differences between “statistical literacy, reasoning and thinking”; research on building students’ reasoning skills about distribution, center and variability; and implications for future research on teaching statistics.
Garfield, J., & Ben-Zvi, D. (2008) Developing students’ statistical reasoning: Connecting research and teaching practice. New York: Springer.
Chapter 13 of this book is titled “Learning to reason about statistical inference.” The chapter discussed the importance of understanding statistical inference, its place in the curriculum, difficulties and misconceptions, teacher’s understanding, students’ dispositions and implications from research. The chapter also identifies instructional tasks that foster development of skills in statistical inference and the ideas and concepts that students should gain through engagement with the tasks. The chapter supports the use of resampling tasks to teach informal inference as well as probability simulations. Finally, the chapter provides suggestions on teaching statistical inference.
Marzano, R. J. (2010). Teaching inference. Educational Leadership, 67(7), 80-81.
In this article, Marzano argues that the process of inference is critical for students to engage in as they develop their critical thinking and reasoning skills. He provides key questions for teachers to ask to elicit inferential responses from their students and promote critical thinking in the classroom. The first question, “What is my inference?”, is designed to help students identify what conclusions or predictions they have made in their statistical process. The second question, “What information did I use to make this inference?”, attempts to help students realize what information they used in their reasoning toward their conclusions. The third question, “How good is my thinking?”, students are guided to analyze whether their process and investigation are valid and that their conclusions are based on strong evidence. The final question Marzano encourages teachers to pose is “Do I need to change my thinking?”, which is designed to build students’ critical thinking skills and attention to new information that may require revision of their original inference. In summary, Marzano presents tools for teachers to help their students develop their skills in inference by asking targeted questions and engaging in critical analysis of their own reasoning.
Paparistodemou, E., & Meletiou-Mavrotheris, M. (2008). Developing young students’ informal inference skills in data analysis. Statistics Education Research Journal. 7(2). 83-106.
In this paper, the authors describe a study where third graders created a survey for students at their school to collect and analyze relevant data. The study suggests that by using data that is interesting and relatable to students, inferential reasoning can begin to develop. The authors also found that using TinkerPlots further enhances students’ development of inferential reasoning by providing an easy way to dynamically view the data and test conjectures. Students in this study were highly motivated and, interestingly, began making data-based arguments on their own. After providing initial support for a conjecture, researchers prompted the students to consider applying the conjecture to a larger population and begin thinking about the ideas of chance. The authors encourage early statistical instruction to be focused on an open-ended approach driven by data where students can begin to reason informally rather than through teaching procedures related to inference. By focusing on an informal approach, the authors argue that students will have time to think through the data and make sense of what is happening in context, thus laying the foundation for a better understanding of formal inferential reasoning later.
Rossman, A. J. (2008). Reasoning about informal statistical inference: One statistician’s view. Statistics Education Research Journal. 7(2). 5-19.
In this article, Rossman contends that statistical inference is what is done after data has been collected. Inference involves the interpretations, implications and insights that follow observations about the data. Rossman presents a comparison between formal and informal inference teaching models and provides a number of examples of activities (specifically randomization tests) to illustrate how inference can be taught in the classroom. Rossman argues that using probability and simulations (repeated sampling, randomization tests) is a powerful way to introduce students to engage in statistical inference and reasoning.
Rossman, A. J., & Chance, B. L. (1999). Teaching the reasoning of statistical inference: A "top ten" list. The College Mathematics Journal. 30(4), 297-305.
In this article, Rossman and Chance argue that critical principles of instruction that include the promotion of “active learning” and work with the practical application of statistics are missing from current instruction of statistical inference. They offer a list of ten key practices that teachers should engage in when teaching statistical inference. For each of these practices, the authors provide examples to illustrate the principles underlying each practice. In order to help students develop inferential reasoning, the Rossman and Chance encourage teachers to provide opportunities for students to perform physical simulations to “make sense” of the data they collect, use technology to enhance exploration of phenomena, and use robust statistical measures, such as tests for significance and confidence intervals, to ensure strength of their interpretations of the data. Rossman and Chance also encourage teachers to emphasize that certain representations and methods of collection can misrepresent the data and result in misinterpretations and false conclusions. Finally, the authors strongly encourage teachers to provide students with frequent opportunities to make connections between their data and the real world and communicate their reasoning through discussion and in written form.
Stohl, H., & Tarr. J. E. (2002). Developing notions of inference using probability simulation tools. Journal of Mathematical Behavior, 21. 319-337.
This article describes the interactions between two sixth-grade students during a twelve day teaching experiment where the students used Probability Explorer software. Six problem-based tasks were used during the teaching experiment to develop students’ reasoning of probability and concept of inference. The tasks required students to model situations, collect, display and analyze data, and generate convincing arguments. As students progressed through the tasks, they developed a foundation for understanding the power of larger samples. The authors also found that interactions with microworld tools were crucial in developing students’ reasoning and combating their misconceptions.
Tarr, J. E., Stohl Lee, H., & Rider, R. L. (2006). When data and chance collide: drawing inference from empirical data. In G. F. Burrill & P. C. Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 139 – 150). Reston, VA: National Council of Teachers of Mathematics.
This chapter discusses a task used with sixth-grade students where they learned to draw inferences from empirical data. The task was called “Schoolopoly”, and students were assigned to test if the dice they were using were “fair” and support their decision with compelling evidence. The task was completed using Probability Explorer, TI graphing calculator or by hand. In summary, students had rich discussions that supported the idea that sample size plays a critical role in statistical inference.
Watson, J. M. (2001). Longitudinal development of inferential reasoning by school students. Educational Studies in Mathematics, 47(3). 337-372.
This article describes follow-up interviews of students in Australia who were involved in the author's previous study (Watson, J.M. & Moritz, J. B., 1999) where students were asked to compare two data sets in graphical form. The author found that in the later interviews, the increase in the outcome of the students’ responses may have been due to developmental changes rather than explicit teaching related to these concepts. Thus, the author reminds teachers that some students may need more support when dealing with analyzing unequal data sets, while other students may be ready to move forward. The author’s findings in these interviews suggest that students in higher grade levels are developmentally more equipped to provide more sophisticated conjectures.
Watson, J. M., & Moritz, J. B. (1999) The beginning of statistical inference: Comparing two data sets. Educational Studies in Mathematics. 37(2), 145-168.
This article describes a study of third to ninth grade students in Australia who were interviewed to assess their understanding when comparing two data sets in graphical form. There were four data sets presented, one set at a time, each increasing in difficulty. As the students reasoned through the problem, interviewers took notes and ranked their responses on a scale describing the level of the students’ responses. They found, that in general, students’ in higher grade levels gave more meaningful responses and descriptions of the data.
Wild, C. J., Pfannkuch, M., Regan, M., & Horton, N. J. (2011). Towards more accessible conceptions of statistical inference. Journal of the Royal Statistical Society. Series A, 174, 247-295.
This article presents a case for intentionally laying a foundation for statistical inference using visual clues. Through computer animations simulating repeated sampling, students can begin to visualize the effects sample size have on variation and, thus, begin to make connections needed to make comparisons leading to inference. The idea is that students will better understand measures of center and can start to make informal inferences by looking at the spread of data as depicted on a box plot. As students progress through the author’s “milestones”, they continue the progression towards better understanding how to make formal inferences. The authors argue that this less complex and concrete foundation is essential for students to begin truly understanding statistical inference.
Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2008). A framework to support research on informal inferential reasoning. Statistics Education Research Journal. 7(2), 40-58.
This article synthesizes research of how informal knowledge and informal reasoning has been defined and how they relate to drawing inferences in statistics. Based on the research, the authors propose a working definition of Inferential Informal Reasoning (IIR) and seek to provide a framework which can help design and evaluate tasks that elicit IIR. Their hope is that this framework will inform and inspire further research and a refined definition of IIR in the future.